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%\author{王立庆（2019级数学与应用数学1班）}
\author{学号 \underline{\hspace{4cm}} 姓名  \underline{\hspace{4cm}} }
%\title{高等代数第六章：向量空间}
\title{第六章向量空间考试}
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\date{2023年3月28日}

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%第七课：数域的概念，向量空间和子空间的概念、例子、基本性质。
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%第八课：向量组的线性相关、线性无关、极大线性无关组，向量组的等价，向量组的秩。
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%第九课：向量空间的基和维数，两个基之间的过渡矩阵，向量的坐标，维数定理。
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%第十课：向量空间之间的同构，矩阵的行空间、列空间、零空间，矩阵的秩，基础解系。
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\begin{enumerate}

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\item %1
设 $V$ 是数域 $F$ 上的向量空间。设 $\alpha\in V$. 设 $\beta_1$ 和 $\beta_2$ 都是 $\alpha$ 的负向量。根据向量空间的公理，证明 $\beta_1=\beta_2$. 

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\item %2
设 $V$ 是数域 $F$ 上的向量空间。设 $U$ 与 $W$ 是 $V$ 的两个向量子空间。记 
$$S = \{ \alpha-\beta \mid \alpha\in U, \beta\in W  \}.$$
证明 $S$ 也是 $V$ 的向量子空间。

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\item %3
设向量组 $\{\alpha_1,\alpha_2,\cdots,\alpha_r\}$ 线性无关，而向量组 $\{\alpha_1,\alpha_2,\cdots,\alpha_r, \beta\}$ 线性相关。
证明向量 $\beta$ 可以由向量组 $\{\alpha_1,\alpha_2,\cdots,\alpha_r\}$ 线性表示。

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\item %4
记 $V=\mathbb{R}[x]$ 为实系数多项式全体组成的向量空间。求下述向量组生成的子空间的维数，
$$\{x+1, x^2+x, x^3+x^2,x^4+x^3\}.$$

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\item %5
证明向量组 $\{ (1,2,0), (2,3,0), (1,1,1) \}$ 是 $V=\mathbb{R}^3$ 的一个基，并求向量 $\xi = (5,6,7)$ 关于这个基的坐标。


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\item %6
设 $V$ 是数域 $F$ 上的 $n$ 维向量空间。证明 $V$ 与 $F^n$ 同构。

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\item %7
矩阵的列空间是指它的列向量组生成的向量子空间。设矩阵 $B$ 可逆。证明矩阵 $A$ 与 $AB$ 有相同的列空间。

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\end{enumerate}



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